bibkey: neumann2009on authors: Walter D. Neumann year: 2009 title: On Leighton’s graph covering theorem doi: 10.48550/arXiv.0906.2496 claim: “Two finite connected regular graphs of the same degree have a common finite cover; the regular case is attributed to Angluin and Gardiner.” strata_touched:

• D5/S3/ConceptDynamics/GraphColoring/GraphCoverDomination license: citation-only triage: anchor

Classical existence mechanism

The introduction and its proof of the regular case in https://arxiv.org/html/0906.2496v2 (2010-05-25) were read on 2026-09-07 Asia/Singapore. The original arXiv posting year is 2009. The text states Leighton’s common finite covering theorem and explicitly attributes its regular-graph special case to Angluin and Gardiner (1981).

Consequence explained here, not quoted as Neumann’s stated theorem: a connected finite d-regular simple graph and K_(d+1) have a common finite cover. The inverse image of one vertex of K_(d+1) is a perfect code, because each closed neighborhood maps bijectively onto K_(d+1). Thus covers of regular graphs admitting perfect codes are already available from classical mathematics. This existence mechanism is not a new contribution of the Annor refutation. The repository supplies a direct port-matching construction with fold d+1, without importing this theorem.

The arXiv title query for Leighton and graph/covering returned six entries, including arXiv:2507.01839v2; its abstract concerns proofs, group commensurability and virtual retracts, not domination. No complete later literature review or global novelty conclusion is claimed.

Verified locator

DOI: https://doi.org/10.48550/arXiv.0906.2496. Independently resolved and checked on 2026-09-07 Asia/Singapore against https://arxiv.org/html/0906.2496v2 (2010-05-25), the introductory statement and “Proof of the k-regular case.” The text assumes connected graphs and attributes the regular common finite cover result to Angluin and Gardiner. The perfect-code consequence above is repository inference from that classical result, not a quoted theorem of Neumann or a novelty claim.